The ebook Elementary Calculus is based on material originally written by H.J. Keisler. For more information please read the copyright pages.


Problems

In Problems 1-36, test the improper integral for convergence and evaluate when possible.

06_applications_of_the_integral-466.gif

06_applications_of_the_integral-467.gif

06_applications_of_the_integral-468.gif

06_applications_of_the_integral-469.gif

06_applications_of_the_integral-470.gif

06_applications_of_the_integral-471.gif

06_applications_of_the_integral-472.gif

06_applications_of_the_integral-473.gif

06_applications_of_the_integral-474.gif

06_applications_of_the_integral-475.gif

06_applications_of_the_integral-476.gif

06_applications_of_the_integral-477.gif

06_applications_of_the_integral-478.gif

06_applications_of_the_integral-479.gif

06_applications_of_the_integral-480.gif

06_applications_of_the_integral-481.gif

06_applications_of_the_integral-482.gif

06_applications_of_the_integral-483.gif

06_applications_of_the_integral-484.gif

06_applications_of_the_integral-485.gif

06_applications_of_the_integral-486.gif

06_applications_of_the_integral-487.gif

06_applications_of_the_integral-488.gif

06_applications_of_the_integral-489.gif

06_applications_of_the_integral-490.gif

06_applications_of_the_integral-491.gif

06_applications_of_the_integral-492.gif

06_applications_of_the_integral-493.gif

06_applications_of_the_integral-494.gif

06_applications_of_the_integral-495.gif

06_applications_of_the_integral-496.gif

06_applications_of_the_integral-497.gif

06_applications_of_the_integral-498.gif

06_applications_of_the_integral-499.gif

06_applications_of_the_integral-500.gif

06_applications_of_the_integral-501.gif

37            Show that if r is a rational number, the improper integral 06_applications_of_the_integral-502.gif converges when r < 1 and diverges when r > 1.

38            Show that if r is rational, the improper integral 06_applications_of_the_integral-503.gif converges when r > 1 and diverges when r < 1.

39            Find the area of the region under the curve y = 4x-2 from x = 1 to x = x.

40            Find the area of the region under the curve 06_applications_of_the_integral-504.gif from x = ½ to x = 1.

41             Find the area of the region between the curves y = x-1/4 and y = x-1/2 from x = 0 to x = 1.

42            Find the area of the region between the curves y = -x-3 and y = x-2, 1 ≤ x < x.

43            Find the volume of the solid generated by rotating the curve y = l/x, 1 ≤ x < x, about (a) the x-axis, (b) the y-axis.

44            Find the volume of the solid generated by rotating the curve y = x-1/3, 0 < x ≤ 1, about (a) the x-axis, (b) the y-axis.

45            Find the volume of the solid generated by rotating the curve y = x-3/2, 0 < v ≤ 4, about (a) the x-axis, (b) the y-axis.

46            Find the volume generated by rotating the curve y = 4x-3, -∞ < ∞ ≤ -2, about

(a) the x-axis, (b) the y-axis.

47            Find the length of the curve 06_applications_of_the_integral-505.gif from x = 0 to x = 1.

48            Find the length of the curve y = x1/3 - 3/5 x5/3 from x = 0 to x = 1.

49            Find the surface area generated when the curve y = √x - ⅓x√x, 0 ≤ x ≤ 1, is rotated about (a) the x-axis, (b) the y-axis.

50            Do the same for the curve y = x1/3 - 3/5 x5/3,0 ≤ x ≤ 1.

51            (a) Find the surface area generated by rotating the curve y = √x, 0 ≤ x ≤ 1, about

the x-axis.

(b)  Set up an integral for the area generated about the y-axis.

52             Find the surface area generated by rotating the curve y = x2/3, 0 ≤ x ≤ 8, about the x-axis.

53            Find the surface area generated by rotating the curve 06_applications_of_the_integral-506.gif, 0 ≤ x ≤ a, about (a) the x-axis, (b) the y-axis (0 < a ≤ r).

54            The force of gravity between particles of mass m1 and m2 is F = gm1m2/s2 where s is the distance between them. If m1 is held fixed at the origin, find the work done in moving m2 from the point (1,0) all the way out the x-axis.

H 55           Show that the Rectangle and Addition Properties hold for improper integrals.


Last Update: 2006-11-25